Improvements in holographic technology have caused holography to play a major role in modern optical systems. Today, holograms are widely used as diffractive optical elements in a wide variety of applications including high resolution imaging systems, focusing and collimating optics, aspheric element testing, and chromatic aberration correction of refractive elements.
It is currently known to fabricate holograms for diffractive optics by creating interference among coherent light beams on a photographic plate and then developing the photographic plate. See U.S. Pat. No. 4,607,914 to Fienup entitled Optical System Design Techniques Using Holographic Optical Element, U.S. Pat. No. 4,649,351 to Veldkamp et al. entitled Apparatus and Method for Coherently Adding Laser Beams, and U.S. Pat. No. 4,813,762 to Leger et al. entitled Coherent Beam Combining of Lasers Using Microlenses and Diffractive Coupling.
Unfortunately, holograms produced by interference among coherent light beams contain internal features which produce the hologram's interference fringes. These interferometric holograms must typically be produced individually, using precision equipment to produce interference among coherent light beams on a photographic plate. These holograms, therefore, are difficult to mass produce.
In order to overcome the mass production problems with interferometric holograms, Computer Generated Holograms ("CGH") have been developed. CGHs have been fabricated by calculating the desired holographic pattern to perform a particular function and then forming the pattern on a glass or other substrate using photolithographic or other techniques. This technique is described, for example, in U.S. Pat. No. 4,960,311 to Moss et al. entitled Holographic Exposure System For Computer Generated Holograms.
It is also known to design these CGHs using iterative discrete encoding. In iterative discrete encoding, the hologram is divided into a two-dimensional array of rectangular cells. An initial transmittance value for each rectangular cell is chosen and the optimized phase for the hologram is calculated. An iterative optimization process is then used to optimize the transmittance values of the cells. An error function for the hologram is then calculated based upon the image quality. A single cell is changed and the change in the output pattern is computed. The error function is then recalculated. Based upon the change in the error function, the change is either accepted or rejected. The process is iteratively repeated until an acceptable value of the error function is reached which optimizes the image quality. The use of computers is ideal for performing these iterations because of the immense time involved in the optical system calculations.
For examples of the use of the iterative encoding method for CGHs see Japanese Patent 59-50480 to Denki K.K. et al. entitled Checking Device of Reproduced Image of Calculating Hologram and U.S. Pat. No. 4,969,700 to Haines entitled Computer Aided Holography and Holographic Computer Graphics. Iterative encoding is also described in publications entitled Computer-Generated Holograms for Geometric Transformations, Applied Optics, Vol. 23, No. 18, pp. 3099-3104, 1984, by Cederquist and Tai, and Computer-Generated Rainbow Holograms, Applied Optics, Vol. 23, No. 14, pp. 2441-47, 1984, by Leseberg and Bryngdahl. Also, an Iterative Discrete On-Axis ("IDO") encoding method is discussed in the publication entitled Iterative Encoding of High-Efficiency Holograms for Generation of Spot Arrays, Optical Society of America, pp. 479-81, 1989, by co-inventor Feldman et. al. the disclosure of which is hereby incorporated by reference.
A major obstacle to implementing such iterative discrete encoding methods for small f-number optical elements has been the low diffraction efficiency (.eta.) achieved by this method, typically less than 50%. Commercial CGH software packages are currently available, such as CODE V from Optical Research Associates of Pasedena, Calif., that model holograms as a continuous phase profile for high efficiency optical systems. Although theoretically iterative encoding methods may achieve 100% diffraction efficiency, a continuous phase profile is necessary in order to achieve such results. Holograms, however, are typically fabricated using Very Large Scale Integration ("VLSI") fabrication technology which is discrete in nature. Unfortunately, VLSI fabrication technology does not allow fabrication of a continuous phase profile. Therefore, when the holograms are fabricated based on the continuous phase profile, the diffraction efficiency is substantially reduced.
In an effort to overcome this fabrication obstacle, radially symmetric holograms have been developed which use discrete phase levels (N) as an approximation to the continuous phase profile. First, the CGHs are modelled as continuous phase-only holograms in an optical design ray tracing program such as CODE V mentioned above. Subsequently, the continuous phase function is sampled and approximated to the nearest discrete phase level (N). This method has become known as the "Direct Sampling" method.
The maximum number of phase levels (N.sub.max) employed in such a CGH is given by, ##EQU1## where .delta. is the smallest feature size capable of fabrication and T.sub.min is the minimum diffractive grating period. For a collimating or focusing lens, T.sub.min is related to the hologram f-number by, ##EQU2## where the f-number may be defined as the focal length of an optical element over the diameter of that optical element, and .lambda. is the wavelength of the light source.
The diffraction efficiency (.eta.) of a CGH using discrete phase levels may be given by ##EQU3##
The Direct Sampling method for fabricating radially symmetric holograms uses a series of J photolithographic masking and etching steps. Holograms fabricated by this method have a series of circular rings and the difference in the number of phase levels (N) between any two adjacent rings is equal to one (1), except between fringes where the difference is equal to N-1. The relationship between the number of masks (J) and the number of phase levels (N) is given by EQU N=2.sup.J ( 4)
The Direct Sampling method is described in U.S. Pat. No. 4,895,790 by Swanson et al. entitled High-Efficiency, Multilevel, Diffractive Optical Elements.
Unfortunately, these known radially symmetric holograms have a low diffraction efficiency when a small f-number is required due to the small feature size limitations which are inherent in the fabrication procedure. Note that as the CGH f-number decreases, the CGH grating period decreases so that according to equation (1) only a small number of phase levels may be employed. This results in a low diffraction efficiency from equation (3). In addition, equation (3) is only valid if the CGH minimum feature size (.delta.) is much smaller than T.sub.min /N.
In practice, for N&gt;2, the actual CGH diffraction efficiency is significantly less than that given by equation (3) when the CGH minimum feature size (.delta.) is comparable to T.sub.min /N. Although diffraction efficiencies exceeding 90% have been reported for large f-number elements (f-number&gt;30), diffraction efficiencies for small f-number CGHs with the above procedure are typically about 40-50% due to the fabrication feature size limitations. Therefore, there is a continued need for a hologram with a high diffraction efficiency when a small f-number is required.